This is a report about stock indexes in the world markets since 24 February 2020 to 17 June 2022 (data collected since the outbreak of the pandemic in Europe until now). For this report I choose selected:

All of data orgin from the website: www.wsj.com/market-data

For analyzis I performed among others: time series models, multiple regression, cross validation.This methods allows to predict values of stock indexes in the future. The calculations are based on the references at the end of the document.

Dow Jones Industrial Average is:

“a stock index that tracks 30 of the largest U.S. companies. Created in 1896, it is one of the oldest stock indexes, and its performance is widely considered to be a useful indicator of the health of the entire U.S. stock market”.

(source of definition: https://www.fool.com/investing/stock-market/indexes/dow-jones/;

source of data: https://www.wsj.com/market-data/quotes/index/DJIA/historical-prices )

FTSE China A50 is:

“an index for 50 stocks of companies with the highest market capitalisation listed on the Shanghai and Shenzhen stock exchanges”.

(source of definition: https://www.avatrade.com/trading-info/financial-instruments-index/indices/china-a50;

source of data: https://www.wsj.com/market-data/quotes/index/XX/XIN9/historical-prices )

FTSE 100 is:

“an index composed of the 100 largest (by market capitalisation ) companies listed on the London Stock Exchange (LSE)”.

(source of definition: https://www.ii.co.uk/knowledge-centre/quick-guides/before-you-start/what-is-the-ftse-100;

source of data: https://www.wsj.com/market-data/quotes/index/UK/UKX/historical-prices)

NASDAQ is:

“The first electronic stock market listing over 5000 companies. The Nasdaq stock market comprises two separate markets, namely the Nasdaq National Market, which trades large, active securities and the Nasdaq Smallcap Market that trades emerging growth companies”.

(source of definition: https://www.nasdaq.com/glossary/n/nasdaq-stock-market;

source of data: https://www.wsj.com/market-data/quotes/index/NASDAQ/historical-prices)

NIKKEI 225 is:

“the most recognized Japanese stock market index. It comprises Japan’s top 225 companies that are listed on the Tokyo Stock Exchange. The Nikkei Index is considered an important measure of the Japanese stock market and the performance of the Japanese economy.”

(source of definition: https://corporatefinanceinstitute.com/resources/knowledge/trading-investing/nikkei-index/;

source of data: https://www.wsj.com/market-data/quotes/index/JP/NIK/historical-prices)

S&P 500 is:

“a market-capitalization-weighted index of 500 leading publicly traded companies in the U.S. It is not an exact list of the top 500 U.S. companies by market cap because there are other criteria that the index includes.”

(source of definition: https://www.investopedia.com/terms/s/sp500.asp;

source of data: https://www.wsj.com/market-data/quotes/index/SPX/historical-prices)

EURO STOCK 50 :

“represents the performance of the 50 largest companies among the 20 supersectors in terms of free-float market cap in Eurozone countries. The index has a fixed number of components and is part of the STOXX blue-chip index family. The index captures about 60% of the free-float market cap of the EURO STOXX Total Market Index (TMI)”

(source of definition: https://www.stoxx.com/document/Bookmarks/CurrentFactsheets/SX5GT.pdf;

source of data: https://www.wsj.com/market-data/quotes/index/XX/SX5E/historical-pricess)

VIX is:

“based on the prices of options on the S&P 500 Index and is calculated by aggregating weighted prices of the index’s call and put options over a wide range of strike prices.”

(source of definition: https://corporatefinanceinstitute.com/resources/knowledge/trading-investing/vix-volatility-index/;

source of data: https://www.wsj.com/market-data/quotes/index/VIX/historical-prices)

This plot below presents all indexes:

This interactive plot below presents value of EURO STOCK 50:

A time series is an ordered set of measurements taken at regular intervals, an ideal example of which is the stock exchange indexes.

Time series plot for EURO STOCK 50 is as follows:

Time series wad broke down into: seasonal component, trend, and residuals.

We see some seasonal character of our data.

Augmented Dickey-Fuller Test for stationarity confirms that variable EURO STOCK 50 is non-stationary:

## 
##  Augmented Dickey-Fuller Test
## 
## data:  tsSX5Euro
## Dickey-Fuller = -2.3361, Lag order = 3, p-value = 0.4438
## alternative hypothesis: stationary

Then it was done time series evaluation using autocovariance (acf function) and partial autocovariance (pacf function). Autocovariance presents the correlation of the time series with itself shifted by a certain time interval. In turn, partial autovariance is the size of the correlation between the time series and its shift (Lander, 2018, s.404).

This plot shows result of autocovariance:

This plot shows result of partial autocovariance:

The charts confirm the non-stationary nature of the trend. Therefore, a differentiation has to be performed. The number of differentiations was determined using the ndiffs function and amounted to 0.

The arima function showed that the optimal model for the discussed time model will be ARMA (1, 0, 0). The ACF and PACF for the ideal model show the white noise pattern:

This is the result of building ARIMA model(1,0,0):

## Series: tsSX5Euro 
## ARIMA(1,0,0) with drift 
## 
## Coefficients:
##          ar1  intercept   drift
##       0.8939  3550.3984  4.0049
## s.e.  0.0798   127.7783  6.1955
## 
## sigma^2 = 3640:  log likelihood = -159.26
## AIC=326.53   AICc=328.19   BIC=332
## 
## Training set error measures:
##                    ME     RMSE      MAE       MPE     MAPE      MASE      ACF1
## Training set 5.772513 57.12563 50.13811 0.1384377 1.367824 0.2815538 0.0824524

This is prediction based on the ARIMA model forecasting for 24 months with the standard error:

Prediction was also performed with Naive Forecasting Method. This is a result:

## 
## Forecast method: Naive method
## 
## Model Information:
## Call: naive(y = tsSX5Euro) 
## 
## Residual sd: 58.9613 
## 
## Error measures:
##                 ME     RMSE      MAE       MPE     MAPE      MASE       ACF1
## Training set 4.155 58.96128 51.23857 0.1057854 1.395605 0.2877335 0.05458368
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jul 2022         3554.8 3479.238 3630.362 3439.238 3670.362
## Aug 2022         3554.8 3447.939 3661.661 3391.371 3718.229
## Sep 2022         3554.8 3423.923 3685.677 3354.641 3754.959
## Oct 2022         3554.8 3403.676 3705.924 3323.676 3785.924
## Nov 2022         3554.8 3385.838 3723.762 3296.396 3813.204
## Dec 2022         3554.8 3369.712 3739.888 3271.732 3837.868
## Jan 2023         3554.8 3354.882 3754.718 3249.052 3860.548
## Feb 2023         3554.8 3341.079 3768.521 3227.941 3881.659
## Mar 2023         3554.8 3328.114 3781.486 3208.114 3901.486
## Apr 2023         3554.8 3315.852 3793.748 3189.361 3920.239

This is a plot:

Prediction was also performed with Holt’s Trend Method:

## 
## Forecast method: Holt's method
## 
## Model Information:
## Holt's method 
## 
## Call:
##  holt(y = tsSX5Euro, h = 24) 
## 
##   Smoothing parameters:
##     alpha = 0.9999 
##     beta  = 1e-04 
## 
##   Initial states:
##     l = 3543.8215 
##     b = 3.0165 
## 
##   sigma:  65.9243
## 
##      AIC     AICc      BIC 
## 346.2808 348.8895 353.1173 
## 
## Error measures:
##                     ME     RMSE      MAE         MPE     MAPE      MASE
## Training set -2.646898 61.20921 53.00171 -0.08589335 1.451215 0.2976345
##                    ACF1
## Training set 0.06696209
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jul 2022       3557.818 3473.333 3642.304 3428.609 3687.028
## Aug 2022       3560.827 3441.347 3680.308 3378.098 3743.557
## Sep 2022       3563.836 3417.498 3710.174 3340.032 3787.641
## Oct 2022       3566.845 3397.861 3735.829 3308.407 3825.283
## Nov 2022       3569.854 3380.916 3758.792 3280.898 3858.809
## Dec 2022       3572.863 3365.882 3779.843 3256.313 3889.412
## Jan 2023       3575.872 3352.296 3799.447 3233.943 3917.801
## Feb 2023       3578.880 3339.857 3817.904 3213.325 3944.436
## Mar 2023       3581.889 3328.354 3835.424 3194.141 3969.638
## Apr 2023       3584.898 3317.636 3852.161 3176.155 3993.641
## May 2023       3587.907 3307.586 3868.228 3159.193 4016.621
## Jun 2023       3590.916 3298.116 3883.716 3143.116 4038.715
## Jul 2023       3593.925 3289.153 3898.696 3127.817 4060.032
## Aug 2023       3596.934 3280.642 3913.225 3113.207 4080.660
## Sep 2023       3599.942 3272.533 3927.352 3099.213 4100.672
## Oct 2023       3602.951 3264.788 3941.115 3085.775 4120.128
## Nov 2023       3605.960 3257.372 3954.548 3072.840 4139.080
## Dec 2023       3608.969 3250.257 3967.681 3060.366 4157.572
## Jan 2024       3611.978 3243.418 3980.538 3048.314 4175.642
## Feb 2024       3614.987 3236.833 3993.140 3036.651 4193.323
## Mar 2024       3617.996 3230.484 4005.507 3025.348 4210.643
## Apr 2024       3621.004 3224.354 4017.655 3014.380 4227.629
## May 2024       3624.013 3218.428 4029.598 3003.725 4244.302
## Jun 2024       3627.022 3212.693 4041.351 2993.361 4260.683

The predictions made by all methods were compared. MAPE and MAE turned out to be the smallest in the ARIMA model.

Multiple regression model for EURO STOXX 50 Index is as follows:

## 
## Call:
## lm(formula = SX5Euro_Close ~ DJ_Close + FTSE_Close + FTSE100_Close + 
##     NASDAQ_Close + NIKKEI_Close + SP500_Close + VIX_Close, data = all_indexes)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -235.225  -55.620    4.422   61.322  232.643 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -6.076e+02  1.124e+02  -5.407 9.72e-08 ***
## DJ_Close       1.607e-01  1.069e-02  15.027  < 2e-16 ***
## FTSE_Close    -2.881e-02  5.533e-03  -5.207 2.76e-07 ***
## FTSE100_Close  4.110e-01  2.613e-02  15.732  < 2e-16 ***
## NASDAQ_Close   2.787e-01  1.773e-02  15.717  < 2e-16 ***
## NIKKEI_Close  -3.225e-02  4.737e-03  -6.808 2.70e-11 ***
## SP500_Close   -1.493e+00  1.118e-01 -13.351  < 2e-16 ***
## VIX_Close      2.803e+00  7.515e-01   3.730 0.000213 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 85.2 on 527 degrees of freedom
##   (71 observations deleted due to missingness)
## Multiple R-squared:  0.9652, Adjusted R-squared:  0.9647 
## F-statistic:  2086 on 7 and 527 DF,  p-value: < 2.2e-16

The model explains 96% of the variability of EURO STOXX 50 Index, and all the variables used in the model are significant.

## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9938         0.0275 
## Kolmogorov-Smirnov        0.0534         0.0945 
## Cramer-von Mises         43.6626         0.0000 
## Anderson-Darling          1.0561         0.0089 
## -----------------------------------------------

The residuals in model meet the assumptions of the normal distribution, and the result of the K-S test also it confirms (p > 0,05 that is, there is no reason to reject the null hypothesis and the distribution is normal).

The Quantile-Quantile plot also confirms the normal distribution of EURO STOXX 50 Index - the observations are almost perfectly positioned on the straight line (except for the rest).

The Cook’s plot above allows you to explore outliers.

Multiple regression models were also compared. The following regression models were built:

They were visualized using the multiplot function.

## Analysis of Variance Table
## 
## Model 1: SX5Euro_Close ~ DJ_Close + FTSE_Close + FTSE100_Close + NASDAQ_Close + 
##     NIKKEI_Close + SP500_Close + VIX_Close
## Model 2: SX5Euro_Close ~ DJ_Close + FTSE100_Close + NASDAQ_Close + SP500_Close + 
##     VIX_Close
## Model 3: SX5Euro_Close ~ VIX_Close + NASDAQ_Close + SP500_Close
## Model 4: SX5Euro_Close ~ VIX_Close + FTSE100_Close
## Model 5: SX5Euro_Close ~ NIKKEI_Close + FTSE_Close
## Model 6: SX5Euro_Close ~ DJ_Close
##   Res.Df      RSS Df Sum of Sq       F    Pr(>F)    
## 1    527  3825475                                   
## 2    529  4645803 -2   -820328  56.505 < 2.2e-16 ***
## 3    531  7358661 -2  -2712859 186.863 < 2.2e-16 ***
## 4    532 14208455 -1  -6849794 943.632 < 2.2e-16 ***
## 5    532 16145983  0  -1937528                      
## 6    533  6057648 -1  10088335                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The smallest RSS is observed in the case of model 1. This is also confirmed by the results of AIC ( Akaike Information Criterion) and BIC ( Bayesian Information Criterion).

Due to the obsolescence of using ANOVA to test regression models (Lander, 2018, p. 337), a cross-validation with generalized linear models was also performed.

##       Error Adjusted Error model_name
## 1  7484.833       7447.452    modelG1
## 2  8941.738       8912.502    modelG2
## 3 13972.780      13948.380    modelG3
## 4 26829.588      26799.546    modelG4
## 5 30456.661      30425.828    modelG5
## 6 11543.813      11519.025    modelG6

It has been shown once again that the first model is characterized by the lowest error value.

In summary, we see fertility rate and birth rate falling over the years, while death rate and life expectancy are increasing. It is very important to follow demographic changes in countries and individual parts of the world so that governments can react early and shape their policies on public health, pro-family policies, etc.It is impossible to track demographic changes in isolation from the data on armed conflicts and the related migration of people, climate change, and the pandemic situation.

References: